Potentials of the Heun class: The triconfluent case
@article{Batic2015PotentialsOT, title={Potentials of the Heun class: The triconfluent case}, author={David Batic and D. Mills-Howell and Marek Nowakowski}, journal={Journal of Mathematical Physics}, year={2015}, volume={56}, pages={052106} }
We study special classes of potentials for which the one-dimensional (or radial) Schrodinger equation can be reduced to a triconfluent Heun equation by a suitable coordinate transformation together with an additional transformation of the wave function. In particular, we analyze the behaviour of those subclasses of the potential arising when the ordinary differential equation governing the coordinate transformation admits explicit analytic solutions in terms of the radial variable. Furthermore…
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References
SHOWING 1-10 OF 34 REFERENCES
Quasi-exactly solvable singular fractional power potentials emerging from the triconfluent Heun equation
- Mathematics, Physics
- 2002
We consider a triconfluent Heun equation (TCHE) possessing infinitely many Liouvillian solutions and transform it into a radial Schrodinger equation for a general power law potential. From the latter…
Potentials of the Heun class
- Mathematics
- 2013
We review different methods of generating potentials such that the one-dimensional Schrodinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with…
Study of the one-dimensional Schroedinger equation generated from the hypergeometric equation
- Mathematics, Physics
- 1999
Original English Summary. - A systematic method of constructing potentials, for which the one-variable Schroedinger equation can be solved in terms of the hypergeometric (HGM) function, is presented.…
The Schrödinger equation
- Mathematics
- 1998
By making use of an ansatz for the eigenfunction, we obtain the exact solutions to the Schrödinger equation with the anharmonic potential, V (r) = ar 2 + br−4 + cr−6, both in three dimensions and in…
Avoided crossings of the quartic oscillator
- Mathematics
- 1997
The phenomenon of avoided crossings of energy levels in the spectrum of quantum systems is well known. However, being of an exponentially small order it is hard to calculate. In particular, this is…
High Energy Eigenfunctions of One-Dimensional Schrödinger Operators with Polynomial Potentials
- Mathematics
- 2007
For a class of one-dimensional Schrödinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit…
Liouville Transformation and Exactly Solvable Schrodinger Equations
- Physics, Mathematics
- 1997
The present paper discusses the connectionbetween exactly solvable Schrodinger equations and theLiouville transformation. This transformation yields alarge class of exactly solvable potentials,…
Anharmonic Oscillator Equations:Treatment Parallel to Mathieu Equation
- Mathematics
- 2004
The treatment of anharmonic oscillators (including double-wells) by instanton methods is wellknown. The alternative differential equation method is not so wellknown. Here we reformulate the latter…
Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic P\"oschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well
- Mathematics
- 2014
It is shown that the regular-at-infinity solution of the 1D Schrodinger equation with the hyperbolic Poschl-Teller (h-PT) potential with integer parameters is expressible in terms of a n-order Heun…
Exactly Solvable Schrödinger Operators
- Mathematics, Physics
- 2010
We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two…